The system (REF ) will be solved iteratively with a simplified
Newton iteration. This means that in each step of the iteration the
new \(K_{1}\)  and \(K_{2}\)  are obtained by inverting the operator
\(\mathcal {L}\)  only instead of the full Jacobian. The vanishing
boundary conditions for \(\tilde{v}\)  and thus for \(K_{1}\) , \(K_{2}\)  are
imposed as in [1]}: the equations
\(\mathcal {L}K_{i}=F_{i}\) , \(i=1,2\) , are solved by considering only the
components \(1,\ldots ,N-1\)  of the \(K_{i}\) ; this means that we consider the reduced
equations
\(\sum _{m=1}^{N-1}\mathcal {L}_{nm}K_{i,m}=F_{i,n},\quad n=0,\ldots ,N-1,\quad i=1,2,\) 
